Please help with below integral transformation.
\begin{align*} T_n(x) & = \cos(n\cos^{-1} x)\\ w(t) & = (1 - x^2)^{-\frac{1}{2}}\\ \varphi_n(x) & = \int_{-1}^{1} \frac{T_n(t) - T_n(x)}{t - x}w(t)~dt\\ \end{align*} How to prove: $$\int_{-1}^{1} \frac{2tT_n(t) - 2xT_n(x)}{t - x}w(t)~dt = 2x\varphi_n(x)$$
We have $$\int_{-1}^{1}\frac{2tT_{n}\left(t\right)-2xT_{n}\left(x\right)}{t-x}w\left(t\right)dt=\int_{-1}^{1}\frac{2xT_{n}\left(t\right)-2xT_{n}\left(x\right)}{t-x}w\left(t\right)dt$$ can be rewritten $$\int_{-1}^{1}\frac{2tT_{n}\left(t\right)-2xT_{n}\left(x\right)-2xT_{n}\left(t\right)+2xT_{n}\left(x\right)}{t-x}w\left(t\right)dt=0\rightarrow2\int_{-1}^{1}T_{n}\left(t\right)w\left(t\right)dt=0$$ so your equality is true iff $$2\int_{-1}^{1}\frac{\cos\left(n\cos^{-1}\left(t\right)\right)}{\sqrt{1-t^{2}}}dt=0$$ in fact $$2\int_{-1}^{1}\frac{\cos\left(n\cos^{-1}\left(t\right)\right)}{\sqrt{1-t^{2}}}dt=\frac{2\sin\left(\pi n\right)}{n}=0.$$